Holography and trace anomaly: what is the fate of (brane-world) black holes? Roberto Casadio∗
arXiv:hep-th/0302171v4 9 Feb 2004
Dipartimento di Fisica, Universit` a di Bologna, and I.N.F.N., Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy The holographic principle relates (classical) gravitational waves in the bulk to quantum fluctuations and the Weyl anomaly of a conformal field theory on the boundary (the brane). One can thus argue that linear perturbations in the bulk of static black holes located on the brane be related to the Hawking flux and that (brane-world) black holes are therefore unstable. We try to gain some information on such instability from established knowledge of the Hawking radiation on the brane. In this context, the well-known trace anomaly is used as a measure of both the validity of the holographic picture and of the instability for several proposed static brane metrics. In light of the above analysis, we finally consider a time-dependent metric as the (approximate) representation of the late stage of evaporating black holes which is characterized by decreasing Hawking temperature, in qualitative agreement with what is required by energy conservation. PACS numbers: 04.70.-s, 04.70.Bw, 04.50.+h
I.
INTRODUCTION
for very large black holes with M ≫ σ −1 ,
The holographic principle [1], in the form of the AdS-CFT conjecture [2], when applied to the RandallSundrum (RS) brane-world [3], yields a relation between classical gravitational perturbations in the bulk and quantum fluctuations of conformal matter fields on the brane [4]. It was then proposed in Refs. [5, 6] that black hole metrics which solve the bulk equations with brane boundary conditions, and whose central singularities are located on the brane, genuinely correspond to quantum corrected (semiclassical) black holes on the brane, rather than to classical ones. A major consequence of such a conjecture would be that no static black holes exist in the brane-world [7], since semiclassical corrections (approximately described by a conformal field theory – CFT – on the brane) have the form of a flux of energy from the source. That black holes are unstable was already well known in the four-dimensional theory, since the Hawking effect [8] makes such objects evaporate, and the semiclassical Einstein equations should hence contain the backreaction of the evaporation flux on the metric. The novelty is that, from the bulk side (of the AdS-CFT correspondence), one understands the Hawking radiation in terms of classical gravitational waves whose origin is the acceleration the black holes are subject to, living on a non-geodesic hypersurface of the (asymptotically) antide Sitter space-time AdS5 [44]. In Ref. [9] a method was proposed by which static brane metrics, such as the asymptotically flat ones put forward in Refs. [10, 11, 12, 13], can be extended into the bulk [45]. The method makes use of the multipole expansion (with respect to the usual areal radial coordinate r on the brane) and proved particularly well suited
∗ Electronic
address:
[email protected]
(1)
where M is the four-dimensional Arnowitt-Deser-Misner (ADM) mass parameter (in geometrical units) and 3 σ the brane tension (times the five-dimensional gravitational constant). The main result is that the horizon of an astrophysical size black hole has the shape of a “pancake” (see also Ref. [16]) and its area is roughly equal to the four-dimensional expression (in terms of M ). To the extent at which the employed approximation is reliable, no singular behavior in the curvature and Kretschmann scalars in the bulk was found, contrary to the case of the Black String (BS) of Ref. [17]. Possible caveats of this approach have already been thoroughly discussed in Ref. [9]. In particular, the convergence of the multiple expansion on the axis of cylindrical symmetry which extends into the bulk is hard to test and the resulting metrics are not completely reliable thereon. As a consequence, it is hard to determine whether the bulk geometry contains singularities on the axis (while no singularity seems to occur far from it) and whether it is asymptotically AdS away from the brane near the axis (while it appears asymptotically AdS far from the axis). In fact, in recent numerical works, regular brane metrics were shown to develop singularities in the bulk when extended by a different method [18] or problems emerged when trying to describe large black holes in asymptotically AdS bulk [19]. If the conjecture of Refs. [5, 6, 7] is correct, it then follows that the static bulk solutions found in Ref. [9] have singularities on the cylindrical axis (possibly far away from the brane) or, at least, are unstable under linear perturbations (of the metric in the bulk), as it occurs for the BS [20]. In either case, it is likely that such metrics will evolve toward different (more stable but yet unknown) configurations [7]. Since the metric elements in Ref. [9] are expressed as sums of many terms (multipoles) and those in Refs. [18, 19] are expressed only numerically, it is practically impossible to carry out a linear perturbation analysis on such backgrounds. One could otherwise try
2 to use the AdS-CFT correspondence backwards in order to estimate the overall effect of bulk perturbations from the established knowledge of the Hawking radiation on the brane. Some information on the latter can be determined straightforwardly from standard four-dimensional expressions provided the brane metric is given (see, e.g., Ref. [21]). One must then check that such information is consistent with known features of the AdS-CFT correspondence before drawing any conclusion about the bulk stability. In fact, the AdS-CFT correspondence requires some general conditions to hold. First of all, one needs the planar limit of the large N expansion of the CFT, that is a large number of conformal fields [2] N ∼ σ −2 ℓ−2 p ≫1 .
(2)
where, in RS models, the four-dimensional Newton constant 8 π GN = ℓ2p (ℓp being the Planck length) is related to the fundamental gravitational length ℓg in five dimensions by ℓ2p = σ ℓ3g [3, 26]. Eq. (2) is therefore tantamount to ℓg ≫ ℓp and assures that stringy effects are negligible. Moreover, the presence of the brane introduces a normalizable four-dimensional graviton and an ultra-violet (UV) cut-off in the CFT, λUV ∼ σ −1 .
(3) II.
The latter must of course be much shorter than any physical low energy scale λIR of interest, λUV ≪ λIR ,
this suggests that the final evolution of a black hole is really unitary. In the next Section we summarize the general concepts with a particular emphasis on the trace anomaly as derived from the point of view of four-dimensional quantum field theory and its comparison with that predicted by the AdS-CFT correspondence. In Section III, we then apply this formalism to candidate static black holes in order to check the reliability of their holographic picture and stability. Our conclusions first of all support the view given in Refs. [9, 24] that static metrics are a good approximation for astrophysical black holes. Moreover, some of the brane metrics analyzed in Ref. [9, 13] are shown to allow for a closer holographic interpretation and to be more stable than the BS. This suggests that brane-world black holes might evaporate more slowly than they would do in a four-dimensional space-time already for very large ADM masses [46]. Finally, in Section IV we discuss a possible candidate time-dependent metric to estimate the late stage of the evaporation by self-consistently including the trace anomaly in the relevant vacuum equation. We shall adopt the brane metric signature (−, +, +, +) and units with ~ = c = 1. Latin indices i, j, . . . will denote brane coordinates throughout the paper.
(4)
in order for the CFT description of the brane-world to be consistent. A further remark is in order about the relevance of the bulk being asymptotically AdS away from the brane. We have already mentioned that, according to Ref. [9], possible singularities in the bulk (and the corresponding deviations from asymptotic AdS) should be located on the axis. To what extent such singularities restrict the holographic description is hard to tell a priori and a general criterion for the validity of the AdS-CFT correspondence will therefore be given in Section II [see Eq. (12) and the following discussion]. So far, no practical advantage seems to emerge from the holographic description over the standard fourdimensional treatment of the back-reaction problem. There is however a point of attack: As we shall review shortly, in the brane-world a vacuum (brane) solution needs to satisfy just one equation, whose analysis is therefore significantly simpler than the full set of four-dimensional vacuum Einstein equations. This will allow us to attempt an approximate description of the late stage of the evaporation which is in qualitative agreement with earlier studies of black holes as extended objects in the microcanonical picture [22, 23], and is characterized by a black hole temperature which vanishes along with the mass of the black hole. Let us remark that the total energy of the system (black hole plus Hawking radiation) is conserved in the microcanonical ensemble and
GENERAL FRAMEWORK
The five-dimensional Einstein equations in (asymptotically) AdS5 with bulk cosmological constant Λ can be projected onto the brane by making use of the GaussCodazzi relations and Israel’s junction conditions (see Ref. [26] for the details). For the RS case which we consider throughout the paper Λ = −σ 2 ℓ3g /6 [3], so that the brane cosmological constant vanishes and the effective four-dimensional Einstein equations become Gij = ℓ2p τij +
ℓ4p πij + Eij , σ2
(5)
where Gij = Rij − (1/2) R gij is the four-dimensional Einstein tensor, τij is the energy-momentum tensor of matter localized on the brane (there is no matter in the bulk) and 1 1 1 1 πij = − τik τj k + τ τij + gij τkl τ kl − gij τ 2 .(6) 4 12 8 24 Where no matter appears on the brane (τij = 0), the existence of an extra spatial dimension manifests itself in the brane-world only in the form of the non-local source term Eij , which is (minus) the projection of the bulk Weyl tensor on the brane and must be traceless [5, 26]. Vacuum brane metrics therefore satisfy Rij = Eij ⇓ R=0.
(7) (8)
3 Of course, Eq. (8) is a weaker condition than the fourdimensional vacuum equation Rij = 0 and, consequently, Birkhoff’s theorem does not necessarily hold for spherically symmetric vacuum brane metrics. The AdS-CFT correspondence should relate the tensor Eij representing (classical) gravitational waves in the bulk to the expectation value of the (renormalized) energy-momentum tensor of conformal fields on the brane [4]. Let us denote the latter by h Tij i. One should then have E ij ∼ ℓ2p h T ij i .
(9)
Since the left hand side above is traceless, such a correspondence can hold if h T i ≡ h T ii i = 0, that is, if the conformal symmetry is not anomalous. Of course, this requires that the UV cut-off (3) be much shorter than any physical length scale in the system. It also requires a “flat” brane (i.e., the absence of any intrinsic four-dimensional length associated with the background) otherwise the CFT will also be affected by that scale. For a black hole, the gravitational radius rh ∼ M is a natural length scale and one therefore expects that only CFT modes with wavelengths much shorter than rh [47] (and still much larger than σ −1 ) propagate freely. One then finds that the necessary condition (4) is equivalent to Eq. (1), that is a reliable CFT descritpion of the Hawking radiation might be possible only for very large black holes of the kind considered in Ref. [9]. From the point of view of the AdS-CFT correspondence, it is the value of bulk perturbations at the boundary that acts as a source for the CFT fields and can give rise to h T iCFT 6= 0. As a check, one can compare with the trace anomaly induced by the presence of a brane as a boundary of AdS in several theories in which the AdS-CFT applies. Since we are just interested in a fourdimensional brane, the case of relevance to us is that of (N stacked) D3-branes (possibly with R = 0) embedded in AdS5 . For such a configuration one finds the holographic Weyl anomaly [27] 1 1 h T iCFT = 2 2 Rij Rij − R2 , (10) 4 ℓp σ 3 which reproduces the conformal anomaly of the fourdimensional M = 4 superconformal SU (N ) gauge theory in the large N limit (2) and vanishes in a fourdimensional (Ricci flat) vacuum [48]. On the other hand, the trace anomaly of the pertinent four-dimensional field theory, h T i4D = h T ii i, can be evaluated independently. It is given in terms of geometrical quantities as well and numerical coefficients which depend on the matter fields. Further, it does not usually vanish on a curved background (even if it is Ricci flat) because, contrary to h T iCFT , it also contains the Kretschmann scalar Rijkl Rijkl . For example, one finds for nB boson fields (see, e.g. Ref. [21]) h T i4D =
nB Rijkl Rijkl − Rij Rij − 2R . (11) 2 2880 π
The term 2R, which is renormalization dependent, would however vanish according to Eq. (8) but we include such term for future reference (see, in particular, Section IV). It is this non-vanishing trace h T i4D which measures the actual violation of the conformal symmetry on the brane. If h T i4D 6= h T iCFT , one needs more than the AdSCFT correspondence to describe the brane physics for the chosen background. In other words, this inequality can be interpreted as signaling the excitation of other matter fields living on the brane (with τ ≡ τ ii ∼ h T i4D − h T iCFT ). The relative difference with respect to h T iCFT , h T i4D − h T iCFT , (12) ΓCFT ≡ h T iCFT
can then be used to estimate to what extent classical gravitational waves in the bulk determine matter fluctuations on the brane [49]. If ΓCFT ≪ 1, then the AdSCFT conjecture implies that the (quantum) brane and (classical) bulk descriptions of black holes are equivalent. Otherwise, since the holography can just account for that part of the Hawking flux which is responsible for h T iCFT , the ratio ΓCFT is also a measure of the relative strength of brane fluctuations (involving other matter modes) with respect to bulk gravitational waves. From the four-dimensional point of view, the trace anomaly is evidence that one is quantizing matter fields, by means of the background field method, on the “wrong” (i.e. unstable) background metric. One should instead find a background and matter state (both necessarily time-dependent) whose metric and energy-momentum tensor solve all relevant field equations simultaneously. This is the aforementioned back-reaction problem of Hawking radiation, whose solution is still out of grasp after several decades from the discovery of black hole evaporation. The authors of Ref. [6] argue that, because of the AdS-CFT correspondence, the problem of describing a brane-world black hole is just as difficult as the (fourdimensional) back-reaction problem itself. One could go even further and claim that it is at least as difficult, since for an holographic black hole the AdS-CFT should be exact and h T i4D = h T iCFT (i.e., ΓCFT = 0), but realistic black holes might involve more “ingredients”. If however one focuses on the brane description, and just considers Eq. (8), the task will simplify considerably. III.
STATIC BLACK HOLES?
We first want to analyze both the semiclassical stability and holography of candidate static brane-world black holes. They are described by asymptotically flat, spherically symmetric metrics which solve Eq. (8), and can be put in the form ds2 = −N (r) dt2 + A(r) dr2 + r2 dΩ2 , 2
2
2
2
(13)
where dΩ = dθ +sin θ dφ , and for the functions N and A we shall now consider several cases previously appeared
4 in the literature. A.
4D and CFT trace anomalies
The first step is to compute h T ij i, h T i4D and h T iCFT . The ratio ΓCFT will then give a measure of the reliability of the holographic picture. 1.
Black String
The quantity [Kij ] is the jump in the extrinsic curvature of the brane and Tij the source term localized on the brane which, for the case at hand, contains the vacuum energy 3 σ and the Hawking flux. One thus has ℓ2p S 1 S [Kij ] = −σ gij + h Tij i − gij h T i4D . (19) σ 3 Since r > 2 M outside the horizon, the second term in the right hand side above is negligible with respect to the first one everywhere in the space accessible to an external observer if
We take the Schwarzschild line element (corresponding to the BS [17]), 2M 1 =1− , (14) A r as the reference brane metric, whose horizon radius is rh = 2 M . The trace of the energy-momentum tensor for nB boson fields on this background is N=
nB M 2 , (15) 60 π 2 r6 and four-dimensional covariant conservation relates it to those parts of the diagonal components of the energymomentum tensor which do not depend on the quantum state of the radiation [32]. Moreover, in the Unruh vacuum, one also has the flux of outgoing Hawking radiation [21, 32], S
h T i4D =
nB K S h T tt i = S h T tr i = − S h T rt i = − S h T rr i ∼ 2 2 ,(16) M r where K is a dimensionless constant. It is therefore expected that the components of the Ricci tensor of the metric which incorporates the back-reaction of the Hawking radiation have a leading behavior nB ℓ2p . (17) M 2 r2 The above terms, representing a steady flux of radiation, are consistent within the adiabatic approximation for which the background (brane) metric is held static. In fact, the constant K (formally) arises from an integration performed over an infinite interval of (Euclidean) time, so as to include all the poles in the Wightman function of the radiation field which yield the Planckian spectrum (the integral diverges and is divided by the length of the time interval to estimate the average flux per unit time [8]). During this large time, the change in the ADM mass is neglected. In order to solve for the back-reaction, one should instead consider the full time dependence of the metric and black hole source. From the point of view of the bulk space-time, the Hawking flux just modifies the junction equations [33] at the brane. The latter, on account of the orbifold symmetry Z2 , will in general read [26] 1 3 [Kij ] = ℓg Tij − gij T . (18) 3 S
E ij ∼
Mσ≫
ℓp . M
(20)
This shows that the Hawking radiation just gives rise to a small perturbation of the bulk metric for black holes of astrophysical size [for which Eq. (1) holds]. This is assumed in the approach of Ref. [9] (and also in the numerical analysis of Refs. [24]) to justify staticity [50]. However, the holographic Weyl anomaly (10) vanishes for this metric (the ratio ΓCFT → ∞) since Rij = 0, and drawing any conclusion from the AdS-CFT correspondence looks questionable. 2.
Case I
From Refs. [11, 13], we consider the functions 1 − 32M 2M r , N =1− , A= 4 r N 1 − 32M r (1 + 9 η)
(21)
the causal structure of whose geometry was analyzed in details in Ref. [13]. One finds that the non-vanishing Ricci tensor components are given by 4 η M2 3 (2 r − 3 M )2 r2 4ηM =− 3 (2 r − 3 M ) r2 4 η M (r − M ) . = I Rφφ = − 3 (2 r − 3 M )2 r2
I
Rt t =
I
Rrr
I
Rθθ
(22)
Note that the corrections in Eq. (17) that one obtains from the Hawking radiation dominate (in the large r approximation) over those following from the components in Eq. (22). This is expected since the metric (21) does not contain an outgoing flux and is asymptotically flat. However, the corrections in Eq. (22) certainly overcome the Hawking flux in the interval 1≪
r M2 . |η| 2 , M ℓp
(23)
whose extension can be very large for astrophysical black holes (with M ≫ σ −1 & ℓp ) provided η is not infinitesimal [51]. Moreover, the trace of the energy-momentum
5 tensor of a boson field on this four-dimensional background has a leading behavior for large r given by (see Appendix A for more details) η2 S η I h T i4D , (24) h T i4D ≃ 1 + + 3 24 which is of the same order in 1/r as the trace in Eq. (15). The expected trace anomaly from the AdS-CFT correspondence on this background is of the same order as I h T i4D , namely I
h T iCFT
η2 M 2 ∼ 2 2 6 . 6 ℓp σ r
(25)
(26)
which is finite for η 6= 0 and represents a significant improvement over the BS. In particular, one has that I ΓCFT ≃ 0 for √ nB ℓ p σ ≡ Iη , (27) η≃± √ 10 π
where we used ℓp σ ≪ 1. For η ≃ I η one expects that the holographic principle yields a reliable description of such black holes. Note however that the rough estimate nB ∼ N from Eq. (2) would yield |I η| ≃ 0.1 which is significantly larger than the experimental bound |η| . 10−4 from solar system measurements [34].
N =
η+
h T i4D ≃ (1 + η) S h T i4D ,
q 1−
2M r
(1 + η)
1+η
2
II
h T iCFT ∼ I h T iCFT ,
yielding a finite (for η 6= 0) ratio II ΓCFT ≃ 0 for
−1 2M (1 + η) , A= 1− r
4.
2 η (1 + η) M q . η + 1 − 2 rM (1 + η) r3
(32)
Case III
1 2M η M2 , =1− + A r 2 r2
N=
(33)
has the Ricci tensor components III
Rtt =
III
Rrr = − III Rθθ = − III Rφφ =
η M2 . (34) 2 r4
The interval over which such corrections overcome the Hawking flux is now narrower, namely 1≪
p M r , . |η| M ℓp
(35)
but still quite large for astrophysical black holes. The corresponding trace anomaly is given by h T i4D ≃ S h T i4D ,
(36)
to leading order in 1/r (see Appendix A). The AdS-CFT trace anomaly is subleading for this case, namely h T iCFT ≃
η2 M 4 , ℓ2p σ 2 r8
(37)
ΓCFT ∼ nB
ℓ2p σ 2 r2 , η2 M 2
(38)
III
Rrr = −2 II Rθθ = −2 II Rφφ =
ΓCFT ∼ I ΓCFT and
Finally, from Ref. [10], the metric
which yield the non-vanishing Ricci tensor components II
II
(31)
where we again used ℓp σ ≪ 1 and |η| ≪ 1.
III
(28)
(30)
and the conformal anomaly from the AdS-CFT correspondence is
Case II
From Refs. [12, 13], let us now consider the metric described by the functions (for the causal structure see again Ref. [13])
II
η ≃ I η/3 ,
Hence, for small |η|, the ratio 2 2 n ℓ σ B p I ΓCFT ∼ 1 − , 10 η 2 π 2
3.
as well. The trace of the boson energy-momentum tensor is also of the same order in 1/r as that in Eq. (15) (see Appendix A)
and (29)
These are again subleading at large r with respect to the radiation terms in Eq. (17), but of the same order as those of Case I, and the estimate in Eq. (23) applies to this case
III
which is larger than those of cases I and II. This result seems therefore to disfavor such a metric as a candidate holographic black hole.
6 B.
Semiclassical stability 1.0001
Since the trace h T i4D 6= 0 signifies that the chosen background is not the true vacuum (for which one would rather expect a vanishing conformal anomaly and toward which the system will evolve), a quantitative way of estimating the stability of the above solutions with respect to the BS is to evaluate the ratio h T i Γ4D ≡ S 4D . (39) hT i
In regions where Γ4D ≪ 1, the corresponding metric should be more stable than the BS. This occurs, for instance, for the candidate small black hole metrics which we shall analyze in Section III B 2 [see Eqs. (46)]. Such brane metrics violate the condition (4) and are therefore unlikely to admit an holographic description, as our approach will indeed confirm [52]. 1.
Cases I, II and III
It is interesting to take note of the approximate asymptotic values of the ratio Γ4D at large r for the three cases previously discussed: I
Γ4D → 1 +
II
η2 η + 3 24
Γ4D → 1 + η
III
(40)
Γ4D
1
0.9999
1e+08
1.5e+08
2e+08
r
FIG. 1: The ratios I Γ4D (solid line) and II Γ4D (dashed line) for η = −10−4 and M = 107 σ −1 (r is in units of σ −1 ).
this background is zero [53], and confronting with the BS (corresponding to n = 1) becomes more subtle. Let us anyways assume rh ∼ M(n) holds from Newtonian arguments (at least for n = 2 [35, 36]), where the M(n) ’s are now understood as the multipole moments of the energy distribution of the source. One then obtains n n (n) t R t = (n) Rrr = − (n) Rθθ = − (n) Rφφ 2 2 n n (n − 1) M(n) = . (42) 2 r2+n Note that the scalar
Γ4D → 1 .
From such expressions, it appears that the preferred solutions are again given by cases I and II, but with η < 0 (in Refs. [9, 13] we already discussed some reasons why one expects η < 0 and the above results further support this conclusion since η > 0 always leads to a larger value of the trace than η ≤ 0). Case III instead represents no real improvement over the BS. In details, the ratios I Γ4D and II Γ4D are plotted in Fig. 1 for the typical values M = 107 σ −1 ≃ 1 km and η = −10−4 [9]. Except for the region very near the horizon (rh = 2 M ), the ratios II Γ4D < I Γ4D < 1. This might signal a stronger instability near the horizon than for the BS which, however, becomes milder at larger distances.
(n)
Small black holes
There are more candidate metrics for small black holes with M σ . 1 (see, e.g., Refs. [16, 36, 37]), namely the higher-dimensional Schwarzschild metrics [38] r n 1 h N= , (41) =1− A r with n ≥ 2. Unfortunately one cannot rigorously identify rh ≃ 2 M , since the four-dimensional ADM mass for
R = (n − 1) (n − 2)
n M(n)
(43)
r2+n
just vanishes for n = 1 (four-dimensional Schwarzschild) and n = 2 (five-dimensional Schwarzschild). These are the only cases which satisfy the vacuum Eq. (8). The complete expression for the trace anomaly is given in Eq. (A1d) which shows that, for n = 2, one has (2)
h T i4D = nB
4 13 M(2)
720 π 2 r8
,
(44)
while, for n > 2, the leading behavior at large r is given by the 2R term, that is (n>2)
2.
5e+07
h T i4D ∼ nB
n (n2 − 1) (n2 − 4) M(n)
5760 π 2 r4+n
,
(45)
both of which never vanish. However, and although the numerical coefficient is questionable because of the aforementioned ambiguity in relating M(n) to rh , the ratios M(2) 2 (2) Γ4D ∼ r (46) n−2 M(n) (n>2) Γ4D ∼ , r
7 10
ρ
which is an increasing function of n and diverges for r → ∞ as occurred for case III. It thus seems that, although such brane metrics are semiclassically more stable, they significantly depart from the holographic description for increasing n. This is not contradictory, since the condition (4) [or, equivalently, Eq. (1)] is now violated and one expects that the CFT description fails. Moreover, one also expects that the smaller the black hole (horizon), the finer the space-time structure is probed, and one eventually needs to include stringy effects.
2 8
6
4
2
1
0.5
1.5
z
2
IV. FIG. 2: The function ρ2 for M(2) = σ −1 and r = M(2) /2, M(2) , 2 M(2) and 3 M(2) . Terms up to order 1/r 13 are included.
tend to zero at large r and are less than one for n > 1 and r & M(n) . This makes such metrics better candidates to describe very small black holes on the brane. It is then interesting to study their extension in the bulk by applying the method of Ref. [9]. Details are given in Appendix B for n = 2 (see also Ref. [40]). For M(2) σ = 1 the value of ρ2 along geodesic lines of constant r is displayed in Fig. 2 and the horizon is approximately given by the line r = M(2) . Note that it is slightly flattened since the maximum value of z along such a line is about 0.7 σ −1 . It would however depart more and more from that curve the smaller M(2) σ is. For larger values of M σ, one expects a non-vanishing ADM mass (2 M = M(1) ), and the line r = rh is then flatter and a better approximation of the true location for the horizon (see Ref. [9] for the cases I, II and III). The trace anomaly from the AdS-CFT correspondence is given in Eq. (A2d), and just vanishes for n = 1. Neglecting numerical coefficients, one thus obtains, for n = 2, a ratio (2)
ΓCFT ≃ 1 − nB
13 ℓ2p σ 2 , 720 π 2
(47)
which vanishes for nB ≃
720 π 2 ≫1, 13 ℓ2p σ 2
(48)
where the inequality follows from the condition (2). For such a number of boson fields one has (2)
h T i4D ≃
4 M(2)
ℓ2p σ 2 r8
.
(49)
Solving Einstein equations for time-dependent metrics is in general a formidable task, unless symmetries are imposed to freeze enough degrees of freedom, and there is little hope to find analytic solutions for the present case [54]. Let us then begin with a qualitative remark based on the results we have shown in the previous Section: Just looking at the trace anomaly (15) one is led to conclude that the natural time evolution of a fourdimensional black hole is toward smaller and smaller ADM masses, since S h T i4D = 0 for M = 0. However, such an evolution seems to make the black hole less and less stable, its specific heat being more and more negative and the temperature diverging, as is argued in the usual canonical picture of the Hawking radiation [8]. But the picture might change if one considers a fully dynamical description (for a recent four-dimensional analysis of Hawking evaporation and trace anomaly, see Ref. [41]). In order to substantiate our argument, let us replace Eq. (8) with the semiclassical brane equation R = −ℓ2p h T i4D ,
ΓCFT ∼
r M(n)
n
,
(50)
(51)
and note that, for a metric of the form (41) and timeindependent rh , the scalar curvature R always falls off more slowly at large r than the corresponding trace anomaly (A1d). Hence, Eq. (51) cannot be solved by such an ansatz . However, the situation changes when the metric is time-dependent: for an asymptotically flat brane metric, on expanding to leading order at large r, the term 2R becomes of the same leading order as R and dominates in the expression of the trace anomaly [55]. In particular, Eq. (51) becomes ¨, R ∼ nB ℓ2p R
(52)
where a dot denotes the derivative with respect to t. We then reconsider the Schwarzschild metric (14) with the simple expression for the ADM mass M = M0 e−a t ,
For n > 2, instead (n>2)
TIME-DEPENDENT BLACK HOLES: A CONJECTURE
(53)
where a > 0 so as to enforce decreasing mass. The Ricci scalar and the trace anomaly for this metric, to leading
8 order at large r, are given by 2 2 −a t M0 −2 a t M0 R = 2a e (54) +O e r r2 2 nB a4 e−a t M0 −2 a t M0 , (55) + O e h T i4D = − 1440 π 2 r r2 and Eq. (51) is thus solved to leading order at large r for √ 2880 π a= √ . (56) nB ℓ p The CFT trace anomaly in this case is subleading, h T iCFT ≃
a4 e−2 a t M02 , 6 ℓ2p σ 2 r2
(57)
and one then concludes that the AdS-CFT correspondence is wildly violated. As we mentioned at the end of Section III B 2, this is not necessarily a flaw. Finally, the non-vanishing components of the energymomentum tensor, again to leading order at large r, are given by T tr ≃ −T rt ≃ T θθ ≃
T φφ
2 a e−a t M ℓ2p r2 (58)
a3 e−a t M , ≃ ℓ2p r
and the luminosity is M˙ ∼ −a M0 e−a t .
(59)
In order to fix a reasonable value for M0 , we can now assume that, for sufficiently large ADM mass, the standard Hawking relation holds [8], M˙ ∼ −nB K
ℓ2p , M2
(60)
where K is the same coefficient which appears in Eq. (16). The transition to the new regime would then occur when the two expression of the luminosity, Eqs. (59) and (60), match (at t = 0), that is for M0 ∼
ℓ2p nB a
!1/3
1/2
≃ 0.1 nB ℓp
≃ 0.1 σ −1 . 0.1 mm ,
(61)
where we have estimated nB as in Eq. (2) in the second line. Since σ M0 < 1, it is no more a surprise that the holographic description fails for the present case. Note that the luminosity (59) vanishes for M = 0 [whereas the expression in Eq. (60) diverges], that is the temperature of such black holes is much lower than the canonical one. This is just the kind of improvement one expects
from energy conservation and the use of the microcanonical picture for the system consisting of the black hole and its Hawking radiation [22] near the end-point of the evaporation [23] [56]. Of course, the above calculations are just suggestive of how to tackle the problem, and are not meant to be conclusive. One point is however clear, that in a braneworld scenario one has to accommodate just for the one vacuum condition in Eq. (51), which is therefore easier to approach than the four-dimensional analogue. The hard part of the task is then moved to the bulk: the brane metric we have considered must not give rise to spurious singularities off the brane. If the evaporation is complete, this is obviously true for the Schwarzschild metric and (53) in the limit t → ∞, but a complete analysis of the bulk equations for such a time-dependent brane metric is intractable analytically. Let us finally speculate on the basis that a vanishing four-dimensional ADM mass is not equivalent to zero proper mass, since terms of higher order in 1/r, such as those considered in Eq. (41), may survive after the time when M has vanished (or, rather, approached the critical value ℓg ). They are in general expected to appear as generated by the non-vanishing Eij and the trace anomaly (A1d) they give rise to is smaller for larger n (and the same value of the “mass” parameter M(n) ). This opens up a wealth of new possibilities for the brane-world. Since we have shown evidence that the late stage of the evaporation is likely a (slow) exponential decay, one can capture an instantaneous picture of its evolution in time [i.e., apply the adiabatic approximation in order to obtain the static form (13)] and expand that in powers of 1/r, X M(n) n N =1− , (62) r n=¯ n
where M(1) ≡ 2 M and n = n ¯ is the smallest order for which the coefficient M(n) 6= 0, so that, although the black hole remains five-dimensional, its profile looks like it is higher-dimensional. Then, one would also have a “remnant” trace anomaly which is approximately given by the expression in Eq. (A1d) with n = n ¯ . If n ¯ increases in time, the corresponding trace anomaly decreases in time and the black hole appears as a higher and higher dimensional object from the point of view of an observer restricted on the four-dimensional brane-world. Correspondingly, the space-time (brane) around the singularity looks flatter and flatter [see the Ricci tensor elements in Eq. (42)]. More precisely, once the horizon radius has approached ℓg , a geometric description of the space surrounding the central singularity becomes questionable, and just the large r limit of the metric can be given sense. The latter is practically flat for r > M(¯n) when n ¯ ≥ 2. Acknowledgments
I thank G. Alberghi, F. Bastianelli, C. Germani, B. Harms and L. Mazzacurati for comments and sug-
9 gestions.
and nB = 1. In obvious notation:
APPENDIX A: TRACE ANOMALIES
We display here the complete expressions of the trace anomalies for the static cases I, II and III of Section III,
I
II
h T i4D
h T i4D
−4 2 M 1 − 32M M r 18 (24 + 8 η + η 2 ) − 16 (162 + 81 η + 10 η 2 ) = 25920 π 2 r6 r 3 4 2 2 M 2 M 2 M , (A1a) +12 (486 + 315 η + 49 η ) 2 + 216 (27 + 21 η + 4 η ) 3 + 9 (243 + 216 η + 48 η ) 4 r r r (1 + η)2 M 2 = 240 π 2 r6 − (1 + η)
η+
h T i4D =
M2 = 60 π 2 r6
h T i4D
n M(n)
5760 π 2 r4+n
!−4 "
r 2M 3 2 2 4 + η (9 + η ) + η (12 + 7 η) 1 − (1 + η) 2 r # ! 2 2M M 2 M 1− , (A1b) (1 + η) + 16 (1 + η) r r r2
2M 1− (1 + η) r r
16 + 27 η 2 + 24 η
III
(n)
r
2
M 13 η 2 M 2 1−η , + r 48 r2
2
4
3
(A1c)
2
(n − 1) (n − 4) − (3 n − 8 n − 23 n + 4)
n M(n)
rn
.
(A1d)
From the AdS-CFT correspondence [27] one instead obtains I
η2 M 2 = 2 2 6 6 ℓp σ r
h T iCFT
II
h T iCFT
3M 1− 2r
3 η 2 (1 − η)2 M 2 = 2 ℓ2p σ 2 r6
III
(n)
η+
r
2M 1− (1 + η) r
!−2
,
(A2a)
(A2b)
η2 M 2 , 4 ℓ2p σ 2 r8
(A2c)
2 2n n2 + 8 n + 4 (1 − n) M(n) . 24 ℓ2p σ 2 r2 (2+n)
(A2d)
h T iCFT =
h T iCFT =
−4 8M 2 M2 1− , + 2 3r r
APPENDIX B: SMALL BLACK HOLES
In the approach of Ref. [9] the bulk metric is taken of the form ds2 = −N (r, z) dt2 + A(r, z) dr2 + ρ2 (r, z) dΩ2 + dz 2 .
(B1)
10 For the brane metric in Eq. (41) with n = 2, the computed metric components (to order 1/r6 , for the sake of simplicity) are then given by ( 2 2 M(2) M(2) 2 −σ z 1 − 2 + (1 − eσ z ) 2 4 (B2a) N = e r σ r ) i M2 h (2) 3σz 2 2 2σz 2 2 σz 2 2 6 + σ M(2) − 12 σ z + 4 e 6 + σ M(2) + e − 2 + σ M(2) − 2 e σ 4 r6 ( 2 i M2 h M(2) (2) 2 2 −σ z + (1 − eσ z ) (B2b) 1 + 2 + σ 2 M(2) A = e r σ 2 r4 ) 2 n o M(2) σz 2σ z 3σ z 4 4 σz 2 2 2 (1 − 2 σ z) + 2 e − 3 (1 + e ) σ M(2) − 3 σ M(2) + 2 1 − 6 e + 3 e 3 σ 4 r6 ) ( 2 2 M(2) 4 M(2) 2 2 −σ z σz 2 σz 2σz 3σ z ρ = r e 1 − (1 − e ) 2 4 + 1 − 6 e + 3 e . (B2c) (1 − 2 σ z) + 2 e σ r 3 σ 4 r6
Note that, as we commented upon in Ref. [9], the function ρ2 vanishes for a finite value of z, with r held fixed, and that locates the axis of cylindrical symmetry in the
Gaussian normal reference frame. The latter covers the whole bulk manifold of such black holes, since no crossing occurs between lines of different constant r (see Fig. 2).
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[49]
[50]
[51]
[52] [53] [54] [55]
[56]
cisely yields the brane tension σ [28] (see also Refs. [29]). For the connection between the Euclidean and the Lorentzian versions of the AdS-CFT correspondence, see Refs. [2, 30] One should be very cautious in applying this argument near the horizon, where the role of (trans)-Planckian physics is not clear (see e.g. [31] and References therein). In the following a (sufficiently) large r expansion will always be assumed. The presence of non-vanishing off-diagonal components of the energy-momentum tensor (16) while Kij and Gij are diagonal may seem to invalidate this argument. However, let us recall that any comparisons between different contributions should be better made in terms of scalar quantities, such as K ii and Kij K ij on the one hand, and h T i and h Tij i h T ij i on the other. In so doing, one precisely obtains conditions of the form given in Eq. (20) with numerical coefficients of order 1. For the typical solar mass M ∼ 1 km, and σ −1 . 1 mm, one has M ∼ 1038 ℓp & 107 σ −1 [9]. This makes the upper limit in the interval (23) of the order of 1076 |η| km or larger, which is practically an infinite extension, even if one considers the experimental bound |η| . 10−4 [34]. The general criterion introduced in Section II leads to this expected property [see Eqs. (47) and (50)]. This result, in turn, supports the validity of our approach. This is the reason why such a metric cannot describe astrophysical black holes [39]. There is an interesting exception: the metric (28) has R = 0 for η and M arbitrary functions of the time (we thank S. Kar for pointing this out to us). The precise coefficient in front of this term depends on the renormalization scheme. However, since all other possible contributions to the trace anomaly would still fall off faster then R at large r and we are just interested in a qualitative result, we shall assume such a factor is of order 1. Analogous results have been obtained in two dimensions [42] for dilatonic black holes which satisfy a principle of least curvature [43].
